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arXiv · 1811.01759

A full-discrete exponential Euler approximation of invariant measure for parabolic stochastic partial differential equations

Ziheng Chen·Siqing Gan·Xiaojie Wang
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Abstract

We discrete the ergodic semilinear stochastic partial differential equations in space dimension $d \leq 3$ with additive noise, spatially by a spectral Galerkin method and temporally by an exponential Euler scheme. It is shown that both the spatial semi-discretization and the spatio-temporal full discretization are ergodic. Further, convergence orders of the numerical invariant measures, depending on the regularity of noise, are recovered based on an easy time-independent weak error analysis without relying on Malliavin calculus. To be precise, the convergence order is $1-\epsilon$ in space and $\frac{1}{2}-\epsilon$ in time for the space-time white noise case and $2-\epsilon$ in space and $1-\epsilon$ in time for the trace class noise case in space dimension $d = 1$, with arbitrarily small $\epsilon>0$. Numerical results are finally reported to confirm these theoretical findings.

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Ziheng Chen, Siqing Gan, Xiaojie Wang. 2018-11-05. A full-discrete exponential Euler approximation of invariant measure for parabolic stochastic partial differential equations. https://doi.org/10.1016/j.apnum.2020.05.008

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